PARTICLE IN A RECTANGULAR BOX
There are only very few problems that can be solved analytically, that is without the help of a computer, just using the right mathematics. One of them that holds an archetypal position in quantum mechanics is the particle in a box. Traditionally it has been used to exemplify and make concrete all the quantum mechanical notions that we have presented so far. However, with the advent of nanoelectronics this simple problem has found practical applications apart from being an archetypal problem of quantum mechanics. We describe it in various approximations immediately below. We begin with the infinite depth quantum well in one dimension.
The well extends from x = 0 to x = L, (see figure 1.3), i.e. the potential energy is zero for 0 ^ x ^ L and infinite otherwise. We consider that an electron can not escape the well because it takes infinite energy to do so. Therefore, the boundary conditions of our problem are
FIGURE 1.3 An infinite quantum well: a finite region where the potential V is zero inside, with infinite potential walls bounding it.
Inside the well, Schroedingers equation in one dimension takes the form The general solution of this equation is
For the boundary condition at x = 0 to hold, the B„ must be zero (because cos(0) = l). Therefore
From the other condition at x = L we get
We are left with only one set of unknowns, the A„. Since VF is a probability we always have to satisfy the normalization condition, equation 1.13, so we have
After some very simple algebra we get
If we substitute 1.17a—1.19 back into the Schroedinger equation, we get or using 1.18
The first three wavefunctions TV TV T*3 together with their modulus squared are shown in figure 1.4. It can be seen that as n increases the nodes of the corresponding wavefunction VP„ increase accordingly. The electron in state n can be found anywhere with probability |VF„|2 but obviously never on the nodes. The most probable position for the nth state is
which looks very plausible.
FIGURE 1.4 The first three eigenstates of an infinite quantum well.
As we can see from the results so far, instead of classical particles the electrons in a quantum well - box behave as oscillating strings i.e. waves and the greatest manifestation of their wave nature is in fact the discretness of their energies.
The electron being in a zero potential energy state in the quantum box has all its energy as kinetic energy. From equation 1.21, we see that the expression Лк„
is the magnitude of the classical momentum of a particle. The reason for the above emphasis is that we are going to meet this expression again in vectorial form in the next section and in the chapter on the physics of electrons in solids. We now turn our attention to a generalization of the 1-dimensional quantum box to 3 dimensions.
The 3 - dimensional quantum box is shown schematically in figure 1.5. It extends along 0 ^ x ^ Lx, 0 ^ у ^ Ly, 0 ^ z ^ Lz. Outside this region the potential is infinite so that the wavefunction ¥ is zero at the faces of the box.
The corresponding Schroedinger equation becomes
If we assume a product form for the wavefunction
FIGURE 1.5 A 3-dimensional infinite quantum well.
then by the method of separation of variables we can decompose the 3-dimensional equation 1.23 into three 1-dimensional equations which have exactly the form of a 1-dimensional quantum well, i.e.
and Ex, Ey> Ez are the kinetic energies along the x, y, z directions respectively and Tikx, etc., are the respective momenta in these directions.
Since the boundary conditions are the same as in the 1-dimensional case we can follow the same arguments for each direction that led us from equation 1.17 to 1.17a and then to 1.18 so that we finally get
with m, n, l integers (positive or negative). There is no essential new physics here except the appearance of 3 quantum numbers which are associated with the three dimensions of space. We wish to emphasize that this is a general feature and we will always expect to find 3 quantum numbers in a realistic 3-dimensional physical problem such as any of the atoms of the periodic table. Every atom behaves as a 3-dimensional potential well, as we will see in section 1.6.